RICHMOND — Virginia Gov. Terry McAuliffe’s chief of staff left a voice-mail message for a Democrat who was on the verge of quitting the General Assembly in June, saying that the senator’s daughter might get a top state job if he stayed to support the governor’s push to expand Medicaid, according to descriptions from three people who heard the recording.

Then-Sen. Phillip P. Puckett wound up resigning, flipping control of the chamber to Republicans and thwarting McAuliffe’s signature goal of expanding health coverage under the Affordable Care Act.

Puckett’s abrupt exit came amid accusations that Republicans had enticed him to leave with job offers for himself and his daughter, triggering an ongoing federal investigation and inflaming partisan passions in Richmond.

Now a voice-mail message suggests that Puckett fielded a similar overture from Paul Reagan, McAuliffe’s chief of staff.

Paul Reagan, Va. Gov. Terry McAuliffe’s chief of staff, left the voicemail for Phillip P. Puckett in the frenzied days leading up to the senator’s resignation in June. (Steve Helber/AP)

“Mr. Reagan acted on his own to inform the senator that there were other available opportunities for which his daughter might apply,” McAuliffe spokesman Brian Coy said. “No further conversations about this topic ever occurred. No position was ever formally offered.”

Puckett’s resignation came at a time of heightened sensitivity to ethics in Richmond, as former governor Robert F. McDonnell (R) and his wife, Maureen, were about to be tried on federal corruption charges. McAuliffe and other Democrats pounced on his departure as more proof that Richmond needed ethics reform. On the same day last week that he unveiled a new ethics commission, McAuliffe called Puckett “despicable” for seeming to trade his Senate seat for job prospects.

Reagan left the message on Puckett’s cellphone in the frenzied days leading up to the senator’s resignation in June. During the same period, the governor and Sen. Mark R. Warner (D-Va.) also called and tried to dissuade Puckett from leaving.

According to the people who heard the recording, Reagan suggested that the governor could make a top agency appointment available to Puckett’s daughter, Martha Puckett Ketron, who had been seeking a District Court judgeship. Under Senate anti-nepotism rules, she could not be appointed to the bench while her father served. She has been doing the job on a temporary basis, and it is unclear whether she will get the permanent appointment.

“I know there was a lot of frustration with your daughter, not, you know, getting a judgeship or something,” Reagan said, according to the transcript provided to The Post. “If there’s something that we can do for her, I mean, you know, we have a couple of big agencies here that we still need agency heads. We could potentially, potentially, subject to approval of the governor and so forth, you know, the Department of Mines, Minerals and Energy could be available.

“So we would be very eager to accommodate her, if, if that would be helpful in keeping you in the Senate. We, we would basically do anything. We just need you really, we need you for the rest of your term and beyond, but in the immediate future, we need you to help us get this Medicaid deal through and I think we’ve got a way to do it.”

Coy said the governor was not aware of Reagan’s message until Thursday, “shortly after you [The Post] called for comment.” When asked what the governor thought of the message, Coy defended Reagan’s actions, suggested that the prospective job offer was not on par with what Republicans are accused of promising Puckett. Coy said: “The governor has full confidence in Paul Reagan.”

Reagan, reached late Thursday, did not deny that he had left the message but offered no further comment. His use of a potential job offer in his failed bid to keep Puckett in the Senate could have political and perhaps even legal consequences for the administration.

Federal prosecutors are investigating whether Puckett gave up his seat in exchange for a job for himself on the state’s Republican-controlled tobacco commission and a judgeship for Ketron. Puckett has said there was no quid pro quo.

Prosecutors have heard the recording, according to two of the people who described it to The Post and spoke on the condition of anonymity to discuss an ongoing criminal matter. Officials in the U.S. Attorney’s Office for the Western District of Virginia did not respond to requests for comment.

Puckett did not respond to a message seeking comment. His attorney, Thomas J. Bondurant of Gentry Locke Rakes & Moore, declined to comment.

Del. Terry Kilgore (R-Scott), the tobacco commission chairman who was involved in lining up a job for Puckett, declined to comment, as did his attorney, Thomas T. Cullen of Woods Rogers.

Regardless of whether prosecutors take an interest in Reagan’s message, it could be an embarrassment to Democrats, who have blasted Republicans for seeming to lure Puckett to leave with job offers. The accounts of the recording suggest that Reagan was exploring whether to make Puckett a counteroffer to stay.

Coy said that the job prospects Reagan floated to keep Puckett in the Senate were not on par with the GOP’s alleged efforts to lure Puckett out with their own offers. He said that long before potential job offers came into play, Senate Republicans had been trying to force Puckett out of the Senate by insisting that his daughter could not be appointed to the bench while he was a member.

“Any comparison between Mr. Reagan’s concern for the political treatment that Mr. Puckett and his family were receiving and the recent scandals are wholly without merit,” Coy said.

In June, as the House and Senate were mired in a months-long Medicaid standoff that threatened to shut down state government, Puckett unexpectedly stepped down.

The news broke along with word that he was expected to get a job as deputy director of the Virginia Tobacco Indemnification and Community Revitalization Commission — and that his daughter was in line for a judgeship.

Puckett withdrew his name from consideration for the tobacco commission job. He acknowledged at the time that he was resigning, in part, to allow his daughter to serve as a juvenile court judge, but he said it was not because the judgeship was being offered as part of a deal.

Puckett’s service in the Senate was a well-known impediment to his daughter’s judicial appointment because of the Senate’s anti-nepotism policy.

Puckett’s supporters have contended that there was nothing unusual, much less illegal, about factoring political considerations into state appointments. They have noted that in 1997, then-Gov. James S. Gilmore III (R) turned a Democratic senator from Loudoun County, Charles L. Waddell, into his deputy transportation secretary, a move that gave the GOP a 20-to-19 edge in what had been an evenly divided Senate.

Gilmore also appointed a Democrat to head the Department of Conservation and Recreation, creating an opening for a Republican to win a seat in the House.

Those moves upset Democrats but spurred no criminal probes.

Puckett had represented Southwest Virginia since 1998. But Republicans easily won his seat in a special election, solidifying GOP control of the legislature.

Reagan is a highly regarded longtime aide to some of Virginia’s most prominent politicians. He served as chief of staff to then-Sen. James Webb (D-Va.), communications director for Warner when he served as governor, and chief of staff to Rep. James P. Moran (D-Va.).

Jenna Portnoy and Matt Zapotosky contributed to this report.

Clarification: An earlier version of this story used imprecise language to describe Gov Terry McAuliffe’s opinion of his chief of staff’s actions in suggesting a job for former senator Phillip P. Puckett’s daughter. This version has been updated to more accurately reflect the governor’s opinion.